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Theorem subgex 12989
Description: The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)
Assertion
Ref Expression
subgex  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  _V )

Proof of Theorem subgex
Dummy variables  s  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subg 12983 . . 3  |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
2 fveq2 5515 . . . . 5  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
32pweqd 3580 . . . 4  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P ( Base `  G
) )
4 oveq1 5881 . . . . 5  |-  ( w  =  G  ->  (
ws  s )  =  ( Gs  s ) )
54eleq1d 2246 . . . 4  |-  ( w  =  G  ->  (
( ws  s )  e. 
Grp 
<->  ( Gs  s )  e. 
Grp ) )
63, 5rabeqbidv 2732 . . 3  |-  ( w  =  G  ->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }  =  { s  e. 
~P ( Base `  G
)  |  ( Gs  s )  e.  Grp }
)
7 id 19 . . 3  |-  ( G  e.  Grp  ->  G  e.  Grp )
8 basfn 12514 . . . . . 6  |-  Base  Fn  _V
9 elex 2748 . . . . . 6  |-  ( G  e.  Grp  ->  G  e.  _V )
10 funfvex 5532 . . . . . . 7  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
1110funfni 5316 . . . . . 6  |-  ( (
Base  Fn  _V  /\  G  e.  _V )  ->  ( Base `  G )  e. 
_V )
128, 9, 11sylancr 414 . . . . 5  |-  ( G  e.  Grp  ->  ( Base `  G )  e. 
_V )
1312pwexd 4181 . . . 4  |-  ( G  e.  Grp  ->  ~P ( Base `  G )  e.  _V )
14 rabexg 4146 . . . 4  |-  ( ~P ( Base `  G
)  e.  _V  ->  { s  e.  ~P ( Base `  G )  |  ( Gs  s )  e. 
Grp }  e.  _V )
1513, 14syl 14 . . 3  |-  ( G  e.  Grp  ->  { s  e.  ~P ( Base `  G )  |  ( Gs  s )  e.  Grp }  e.  _V )
161, 6, 7, 15fvmptd3 5609 . 2  |-  ( G  e.  Grp  ->  (SubGrp `  G )  =  {
s  e.  ~P ( Base `  G )  |  ( Gs  s )  e. 
Grp } )
1716, 15eqeltrd 2254 1  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   {crab 2459   _Vcvv 2737   ~Pcpw 3575    Fn wfn 5211   ` cfv 5216  (class class class)co 5874   Basecbs 12456   ↾s cress 12457   Grpcgrp 12831  SubGrpcsubg 12980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-inn 8918  df-ndx 12459  df-slot 12460  df-base 12462  df-subg 12983
This theorem is referenced by:  isnsg  13015
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